Brane New World

arXiv:hep-th/0003052 v3 13 Apr 2000

Brane New World

S.W. Hawking

∗

, T. Hertog

†

and H.S. Reall

‡

DAMTP

Centre for Mathematical Sciences

University of Cambridge

Wilberforce Road, Cambridge CB3 0WA, UK.

Preprint DAMTP-2000-25

March 7, 2000

Abstract

We study a Randall-Sundrum cosmological scenario consisting of a domain wall in

anti-de Sitter space with a strongly coupled large N conformal field theory living on the
wall. The AdS/CFT correspondence allows a fully quantum mechanical treatment of this
CFT, in contrast with the usual treatment of matter fields in inflationary cosmology. The
conformal anomaly of the CFT provides an effective tension which leads to a de Sitter
geometry for the domain wall. This is the analogue of Starobinsky’s four dimensional
model of anomaly driven inflation. Studying this model in a Euclidean setting gives a
natural choice of boundary conditions at the horizon. We calculate the graviton correlator
using the Hartle-Hawking “No Boundary” proposal and analytically continue to Lorentzian
signature. We find that the CFT strongly suppresses metric perturbations on all but the
largest angular scales. This is true independently of how the de Sitter geometry arises, i.e.,
it is also true for four dimensional Einstein gravity. Since generic matter would be expected
to behave like a CFT on small scales, our results suggest that tensor perturbations on small
scales are far smaller than predicted by all previous calculations, which have neglected the
effects of matter on tensor perturbations.

Introduction

Randall and Sundrum (RS) have suggested [1] that four dimensional gravity may be recovered
in the presence of an infinite fifth dimension provided that we live on a domain wall embedded

∗

email: S.W.Hawking@damtp.cam.ac.uk

†

Aspirant FWO-Vlaanderen; email: T.Hertog@damtp.cam.ac.uk

‡

email: H.S.Reall@damtp.cam.ac.uk

Figure 1: Carter-Penrose diagram of anti-de Sitter space with a flat domain wall. The dotted
line denotes timelike infinity and the arrows denote identifications. The heavy dots denote
points at infinity. Note that the Cauchy horizons intersect at infinity.

in anti-de Sitter space (AdS). Their linearized analysis showed that there is a massless bound
state of the graviton associated with such a wall as well as a continuum of massive Kaluza-Klein
modes. More recently, linearized analyses have examined the spacetime produced by matter
on the domain wall and concluded that it is in close agreement with four dimensional Einstein
gravity [2, 3].

RS used horospherical coordinates based on slicing AdS into flat hypersurfaces. These horo-

spherical coordinates break down at the horizons shown in figure 1. An issue that has not
received much attention so far is the role of boundary conditions at these Cauchy horizons in
AdS. With stationary perturbations, one can impose the boundary conditions that the horizons
remain regular. Indeed, without this boundary condition the solution for stationary perturba-
tions is not well defined. Even for non-perturbative departures from the RS solution, like black
holes, one can impose the boundary condition that the AdS horizons remain regular [4, 5, 2, 6, 7].
Non-stationary perturbations on the domain wall, however, will give rise to gravitational waves
that cross the horizons. This will tend to focus the null geodesic generators of the horizon,
which will mean that they will intersect each other on some caustic. Beyond the caustic, the
null geodesics will not lie in the horizon. However, null geodesic generators of the future event
horizon cannot have a future endpoint [8] and so the endpoint must lie to the past. We con-
clude that if the past and future horizons remain non-singular when perturbed

(as required for

a well-defined boundary condition) then they must intersect at a finite distance from the wall.
By contrast, the past and future horizons don’t intersect in the RS ground state but go off to
infinity in AdS.

The RS horizons are like the horizons of extreme black holes. When considering perturba-

tions of black holes, one normally assumes that radiation can flow across the future horizon but
that nothing comes out of the past horizon. This is because the past horizon isn’t really there,
and should be replaced by the collapse that formed the black hole. To justify a similar boundary
condition on the Randall-Sundrum past horizon, one needs to consider the initial conditions of
the universe.

The main contender for a theory of initial conditions is the “no boundary” proposal

[10]

It has been shown that the KK modes of RS give rise to singular horizons [9].

Other approaches to quantum cosmology in the RS model have been discussed in [11, 12]. Boundary

Figure 2: Anti-de Sitter space with a de Sitter domain wall. AdS is drawn as a solid cylinder,
with the boundary of the cylinder (dashed line) representing timelike infinity. The light cone
shown is the horizon. The arrows denote identifications.

that the quantum state of the universe is given by a Euclidean path integral over compact
metrics. The simplest way to implement this proposal for the Randall Sundrum idea is to
take the Euclidean version of the wall to be a four sphere at which two balls of AdS

are joined

together. In other words, take two balls in AdS

, and glue them together along their four sphere

boundaries. The result is topologically a five sphere, with a delta function of curvature on a
four dimensional domain wall separating the two hemispheres. If one analytically continues
to Lorentzian signature, one obtains a four dimensional de Sitter hyperboloid, embedded in
Lorentzian anti de Sitter space, as shown in figure 2. The past and future RS horizons, are
replaced by the past and future light cones of the points at the centres of the two balls. Note
that the past and future horizons now intersect each other and are non extreme, which means
they are stable to small perturbations. A perfectly spherical Euclidean domain wall will give
rise to a four dimensional Lorentzian universe that expands forever in an inflationary manner

In order for a spherical domain wall solution to exist, the tension of the wall must be larger

than the value assumed by RS, who had a flat domain wall. We shall assume that matter on the
wall increases its effective tension, permitting a spherical solution. In section 3, we consider a
strongly coupled large N CFT on the domain wall. On a spherical domain wall, the conformal
anomaly of the CFT increases the effective tension of the domain wall, making the spherical
solution possible. The Lorentzian geometry is a de Sitter universe with the conformal anomaly
driving inflation

, an idea introduced long ago by Starobinsky [19].

The no boundary proposal allows one to calculate unambiguously the graviton correlator on

the domain wall. In particular, the Euclidean path integral itself uniquely specifies the allowed
fluctuation modes, because perturbations that have infinite Euclidean action are suppressed
in the path integral. Therefore, in this framework, there is no need to impose by hand an
additional, external prescription for the vacuum state for each perturbation mode. In addition,

conditions motivated by a Euclidean approach were also used in [3] for a flat domain wall.

Such inflationary brane-world solutions have been studied in [13, 14, 15, 16, 11]. For a discussion of other

cosmological aspects of the RS model, see [17] and references therein.

A similar idea was recently discussed within the context of renormalization group flow in the AdS/CFT

correspondence [18]. However, in the case the CFT was the CFT dual to the bulk AdS geometry, not a new
CFT living on the domain wall.

the AdS/CFT correspondence allows a fully quantum mechanical treatment of the CFT, in
contrast with the usual classical treatment of matter fields in inflationary cosmology.

Finally, we analytically continue the Euclidean correlator into the Lorentzian region, where

it describes the spectrum of quantum mechanical vacuum fluctuations of the graviton field on an
inflating domain wall with conformally invariant matter living on it. We find that the quantum
loops of the large N CFT give spacetime a rigidity that strongly suppresses metric fluctuations
on small scales. Since any matter would be expected to behave like a CFT at small scales, this
result probably extends to any inflationary model with sufficiently many matter fields. It has
long been known that matter loops lead to short distance modifications of gravity. Our work
shows that these modifications can lead to observable consequences in an inflationary scenario.

Although we have carried out our calculations for the RS model, we shall show that results

for four dimensional Einstein gravity coupled to the CFT can be recovered by taking the domain
wall to be large compared with the AdS scale. Thus our conclusion that metric fluctuations are
suppressed holds independently of the RS scenario.

The spherical domain wall considered in this paper analytically continues to a Lorentzian

de Sitter universe that inflates forever. However, Starobinsky [19] showed that the conformal
anomaly driven de Sitter phase is unstable to evolution into a matter dominated universe. If
such a solution could be obtained from a Euclidean instanton then it would have an O(4)
symmetry group, rather than the O(5) symmetry of a spherical instanton. We shall study such
models for both the RS model and four dimensional Einstein gravity in a separate paper.

The AdS/CFT correspondence [20, 21, 22] provides an explanation of the RS behaviour

[23]. It relates the RS model to an equivalent four dimensional theory consisting of general
relativity coupled to a strongly interacting conformal field theory and a logarithmic correction.
Under certain circumstances, the effects of the CFT and logarithmic term are negligible and
pure gravity is recovered. We review this correspondence in section 2.

In section 3 we present our calculation of the graviton correlator on the instanton and demon-

strate how the result is continued to Lorentzian signature. Section 4 contains our conclusions
and some speculations. This paper also includes two appendices which contain technical details
that we have omitted from the text.

Randall-Sundrum from AdS/CFT

The AdS/CFT correspondence [20, 21, 22] relates IIB supergravity theory in AdS

× S

to a

N = 4 U(N) superconformal field theory. If g

Y M

is the coupling constant of this theory then

the ’t Hooft parameter is defined to be λ = g

Y M

N . The CFT parameters are related to the

supergravity parameters by [20]

l = λ

1/4

(2.1)

(2.2)

where l

is the string length, l the AdS radius and G the five dimensional Newton constant.

Note that λ and N must be large in order for stringy effects to be small. The CFT lives on the

This was first pointed out in unpublished remarks of Maldacena and Witten.

conformal boundary of AdS

. The correspondence takes the following form:

Z[h]

≡

d[g] exp(

−S

grav

[g]) =

d[φ] exp(

−S

CF T

[φ; h])

≡ exp(−W

CF T

[h]),

(2.3)

here Z[h] denotes the supergravity partition function in AdS

. This is given by a path integral

over all metrics in AdS

which induce a given conformal equivalence class of metrics h on the

conformal boundary of AdS

. The correspondence relates this to the generating functional

CF T

of connected Green’s functions for the CFT on this boundary. This functional is given

by a path integral over the fields of the CFT, denoted schematically by φ. Other fields of the
supergravity theory can be included on the left hand side; these act as sources for operators of
the CFT on the right hand side.

A problem with equation 2.3 as it stands is that the usual gravitational action in AdS is

divergent, rendering the path integral ill-defined. A procedure for solving this problem was
developed in [22, 24, 25, 26, 27, 28, 29]. First one brings the boundary into a finite radius.
Next one adds a finite number of counterterms to the action in order to render it finite as the
boundary is moved back off to infinity. These counterterms can be expressed solely in terms of
the geometry of the boundary. The total gravitational action for AdS

d+1

becomes

grav

= S

+ S

+ . . . .

(2.4)

The first term is the usual Einstein-Hilbert action

with a negative cosmological constant:

−

16πG

d+1

√

R +

d(d

− 1)

(2.5)

the overall minus sign arises because we are considering a Euclidean theory. The second term
in the action is the Gibbons-Hawking boundary term, which is necessary for a well-defined
variational problem [30]:

−

8πG

√

hK,

(2.6)

where K is the trace of the extrinsic curvature of the boundary

and h the determinant of the

induced metric. The first two counterterms are given by the following [26, 27, 28, 29] (we use
the results of [29] rotated to Euclidean signature)

− 1

8πGl

√

(2.7)

16πG(d

− 2)

√

hR,

(2.8)

where R now refers to the Ricci scalar of the boundary metric. The third counterterm is

16πG(d

− 2)

− 4)

√

−

4(d

− 1)

(2.9)

We use a positive signature metric and a curvature convention for which a sphere has positive Ricci scalar.

Our convention is the following. Let n denotes the outward unit normal to the boundary. The extrinsic

curvature is defined as K

µν

= h

∇

, where h

= δ

− n

projects quantities onto the boundary.

where R

is the Ricci tensor of the boundary metric and boundary indices i, j are raised and

lowered with the boundary metric h

. This expression is ill-defined for d = 4, which is the case

of most interest to us. With just the first two counterterms, the gravitational action exhibits
logarithmic divergences [24, 25, 26] so a third term is needed. This term cannot be written
solely in terms of a polynomial in scalar invariants of the induced metric and curvature tensors;
it makes explicit reference to the cut-off (i.e. the finite radius to which the boundary is brought
before taking the limit in which it tends to infinity). The form of this term is the same as 2.9
with the divergent factor of 1/(d

− 4) replaced by log(R/ρ), where R measure the boundary

radius and ρ is some finite renormalization length scale.

Following [23], we can now use the AdS/CFT correspondence to explain the behaviour

discovered by Randall and Sundrum. The (Euclidean) RS model has the following action:

= S

+ S

+ 2S

+ S

(2.10)

Here 2S

is the action of a domain wall with tension (d

−1)/(4πGl). The final term is the action

for any matter present on the domain wall. The domain wall tension can cancel the effect of
the bulk cosmological constant to produce a flat domain wall. However, we are interested in a
spherical domain wall so we assume that the matter on the wall gives an extra contribution to
the effective tension. We shall discuss a specific candidate for the matter on the wall later on.
The wall separates two balls B

and B

of AdS.

We want to study quantum fluctuations of the metric on the domain wall. Let g

denote

the five dimensional background metric we have just described and h

the metric it induces on

the wall. Let h denote a metric perturbation on the wall. If we wish to calculate correlators of
h on the domain wall then we are interested in a path integral of the form

(x)h

)

i =

d[h]Z[h]h

(x)h

(2.11)

where

Z[h] =

∪B

d[δg]d[φ] exp(

−S

+ δg])

= exp(

−2S

+ h])

(2.12)

∪B

d[δg]d[φ] exp(

−S

+ δg]

− S

+ δg]

− S

[φ; h

+ h]),

δg denotes a metric perturbation in the bulk that approaches h on the boundary and φ denotes
the matter fields on the domain wall. The integrals in the two balls are independent so we can
replace the path integral by

Z[h] = exp(

−2S

+ h])

d[δg] exp(

−S

+ δg]

− S

+ δg])

d[φ] exp(

−S

[φ; h

+ h]),

(2.13)

In principle, we should worry about gauge fixing and ghost contributions to the gravitational action. A con-

venient gauge to use in the bulk is transverse traceless gauge. We shall only deal with metric perturbations that
also appear transverse and traceless on the domain wall. The gauge fixing terms vanish for such perturbations
and the ghosts only couple to these perturbations at higher orders.

where B denotes either ball. We now take d = 4 and use the AdS/CFT correspondence 2.3 to
replace the path integral over δg by the generating functional for a conformal field theory:

d[δg] exp(

−S

+ δg]

− S

+ δg]) =

exp(

−W

+ h] + S

+ h]),

(2.14)

we shall refer to this CFT as the RS CFT since it arises as the dual of the RS geometry. It has
gauge group U(N

), where N

is given by equation 2.2. Strictly speaking, we are using an

extended form of the AdS/CFT conjecture, which asserts that supergravity theory in a finite
region of AdS is dual to a CFT on the boundary of that region with an ultraviolet cut-off related
to the radius of the boundary

. The path integral for the metric perturbation becomes

Z[h] = exp(

−2W

+ h] + 2S

+ h])

d[φ] exp(

−S

[φ; h

+ h]). (2.15)

The RS model has been replaced by a CFT and a coupling to matter fields and the domain wall
metric given by the action

− 2S

+ h]

− 2S

+ h] + S

[φ; h

+ h].

(2.16)

The remarkable feature of this expression is that the term

−2S

is precisely the (Euclidean)

Einstein-Hilbert action for four dimensional gravity with a Newton constant given by the RS
value

= G/l.

(2.17)

Therefore the RS model is equivalent to four dimensional gravity coupled to a CFT with cor-
rections to gravity coming from the third counter term. This explains why gravity is trapped
to the domain wall.

At first sight this appears rather amazing. We started off with a quite complicated five

dimensional system and have argued that it is dual to four dimensional Einstein gravity with
some corrections and matter fields. However in order to use this description, we have to know
how to calculate with the RS CFT. At present, the only way we know of doing this is via
AdS/CFT, i.e., going back to the five dimensional description. The point of the AdS/CFT
argument is to explain why the RS “alternative to compactification” works and also to explain
the origin of the corrections to Einstein gravity in the RS model. Note that if the matter on
the domain wall dominates the RS CFT and the third counterterm then these can be neglected
and a purely four dimensional description is adequate.

CFT on the Domain Wall

3.1

Introduction

Long ago, Starobinsky studied the cosmology of a universe containing conformally coupled
matter [19]. CFTs generally exhibit a conformal anomaly when coupled to gravity (for a review,

Evidence in support of this extended version of the duality was given in [31].

see [32]). Starobinsky gave a de Sitter solution in which the anomaly provides the cosmological
constant. By analyzing homogeneous perturbations of this model, he showed that the de Sitter
phase is unstable but could be long lived, eventually decaying to a FRW cosmology.

In this section we will consider the RS analogue of Starobinsky’s model by putting a CFT

on the domain wall. On a spherical domain wall, the conformal anomaly provides the extra
tension required to satisfy the Israel equations. It is appealing to choose the new CFT to
be a

N = 4 superconformal field theory because then the AdS/CFT correspondence makes

calculations relatively easy

. This requires that the CFT is strongly coupled, in contrast with

Starobinsky’s analysis

Our five dimensional (Euclidean) action is the following:

S = S

+ S

+ 2S

+ W

CF T

(3.1)

We seek a solution in which two balls of AdS

are separted by a spherical domain wall. Inside

each ball, the metric can be written

= l

(dy

+ sinh

ydΩ

(3.2)

with 0

≤ y ≤ y

. The domain wall is at y = y

and has radius

R = l sinh y

(3.3)

The effective tension of the domain wall is given by the Israel equations as

ef f

4πGl

coth y

(3.4)

The actual tension of the domain wall is

σ =

4πGl

(3.5)

We therefore need a contribution to the effective tension from the CFT. This is provided by the
conformal anomaly, which takes the value [24, 25, 26]

hT i = −

8π

(3.6)

This contributes an effective tension

−hT i/4. We can now obtain an equation for the radius of

the domain wall:

+ 1 =

8πl

(3.7)

It is easy to see that this has a unique positive solution for R. We shall derive this equation
directly from the action in subsection 3.3.

We emphasize that this use of the AdS/CFT correspondence is independent of the use described above

because this new CFT is unrelated to the RS CFT.

Note that the conformal anomaly is the same at strong and weak coupling [25] so any differences arising

from strong coupling can only show up when we perturb the system.

We are particularly interested in how perturbations of this model would appear to inhabitants

of the domain wall. Thus we are interested in metric perturbations on the sphere

= (R

+ h

)dx

(3.8)

Here ˆ

is the metric on a unit d-sphere. We shall only consider tensor perturbations, for which

is transverse and traceless with respect to ˆ

. In order to calculate correlators of the metric

perturbation, we need to know the action to second order in the perturbation. The most difficult
part here is obtaining W

CF T

to second order. This is the subject of the next subsection.

3.2

CFT Generating Function

We want to work out the effect of the perturbation on the CFT on the sphere. To do this we use
AdS/CFT. Introduce a fictional AdS region that fills in the sphere. Let ¯l, ¯

G be the AdS radius

and Newton constant of this region. We emphasize that this region has nothing to do with the
regions of AdS that “really” lie inside the sphere in the RS scenario. This new AdS region is
bounded by the sphere. If we take ¯l to zero then the sphere is effectively at infinity in AdS so
we can use AdS/CFT to calculate the generating functional of the CFT on the sphere. In other
words, ¯l is acting like a cut-off in the CFT and taking it to zero corresponds to removing the
cut-off. However the relation

¯l

(3.9)

implies that if ¯l is taken to zero then we must also take ¯

G to zero since N is fixed (and large).

For the unperturbed sphere, the metric in the new AdS region is

= ¯l

(dy

+ sinh

yˆ

(3.10)

and the sphere is at y = y

given by R = ¯lsinh y

. Note that y

→ ∞ as ¯l→ 0 since R is fixed.

In order to use AdS/CFT for the perturbed sphere, we need to know how the perturbation
extends into the bulk. This is done by solving the linearized Einstein equations. It is always
possible to choose a gauge in which the bulk metric perturbation takes the form

(y, x)dx

(3.11)

where h

is transverse and traceless with respect to the metric on the spherical spatial sections:

(x)h

(y, x) = ˆ

∇

(y, x) = 0,

(3.12)

with ˆ

∇ denoting the covariant derivative defined by the metric ˆγ

. Since we are only dealing with

tensor perturbations, this choice of gauge is consistent with the boundary sitting at constant y. If
scalar metric perturbations were included then we would have to take account of a perturbation
in the position of the boundary. These issues are discussed in detail in Appendix A.

The linearized Einstein equations in the bulk are (for any dimension)

∇

µν

−

¯l

µν

(3.13)

where µ, ν are d + 1 dimensional indices. It is convenient to expand the metric perturbation in
terms of tensor spherical harmonics H

(p)

(x). These obey

(p)

(x) = ˆ

∇

(p)

(x) = 0,

(3.14)

and they are tensor eigenfunctions of the Laplacian:

∇

(p)

= (2

− p(p + d − 1)) H

(p)

(3.15)

where p = 2, 3, . . .. We have suppressed extra labels k, l, m, . . . on these harmonics. The
harmonics are orthonormal with respect to the obvious inner product. See Appendix B and
[33] for more details of their properties. The metric perturbation can be written as a sum of
separable perturbations of the form

(y, x) = f

(y)H

(p)

(x).

(3.16)

Substituting this into equation 3.13 gives

(y) + (d

− 4) coth yf

(y)

− (2(d − 2) + (p(p + d − 1) + 2(d − 3))cosech

y)f

(y) = 0. (3.17)

The roots of the indicial equation are p + 2 and

−p − d + 3, yielding two linearly independent

solutions for each p. In order to compute the generating functional W

CF T

we have to calculate

the Euclidean action of these solutions. However, because the latter solution goes as y

−(p+d−3)

at the origin y = 0 of the instanton, the corresponding fluctuation modes have infinite Euclidean
action

. Hence they are suppressed in the path integral. Therefore, in contrast to other methods

[2, 3] where one requires a (rather ad hoc) prescription for the vacuum state of each perturbation
mode, there is no need to impose boundary conditions by hand in our approach: the Euclidean
path integral defines its own boundary conditions, which automatically gives a unique Green
function. The path integral unambiguously specifies the allowed fluctuation modes as those
which vanish at y = 0. Note that boundary conditions at the origin in Euclidean space replace
the need for boundary conditions at the horizon in Lorentzian space. The solution regular at
y = 0 is given by

(y) =

sinh

p+2

cosh

F (p/2, (p + 1)/2, p + (d + 1)/2, tanh

y).

(3.18)

This solution can also be written in terms of associated Legendre functions:

(y)

∝ (sinh y)

−d)/2

−(p+(d−1)/2)

−(d+1)/2

(cosh y)

∝ (sinh y)

−d)/2

d/2
p+(d

−2)/2

(coth y),

(3.19)

and the latter can be related to Legendre functions if d/2 is an integer, using

(z) = (z

− 1)

m/2

(3.20)

This can be seen by surrounding the origin by a small sphere y = and calculating the surface terms in the

action that arise on this sphere. They are the same as the surface terms in equations 3.25 and 3.26 below, which
are obviously divergent for the modes in question.

The full solution for the metric perturbation is

(y, x) =

(y)

)

(p)

(x)

γh

(p)

(3.21)

We have a solution for the metric perturbation throughout the bulk region. The AdS/CFT

correspondence can now be used to give the generating functional of the CFT on the perturbed
sphere:

CF T

= S

+ S

+ . . . .

(3.22)

We shall give the terms on the right hand side for d = 4.

The Einstein-Hilbert action with cosmological constant is

−

16π ¯

√

R +

¯l

(3.23)

and perturbing this gives

bulk

−

16π ¯

√

−

¯l

1
4

µν

∇

µν

2¯l

µν

−

16π ¯

√

−

1
2

νρ

∇

µρ

3
4

νρ

∇

νρ

(3.24)

where Greek indices are five dimensional and we are raising and lowering with the unperturbed
five dimensional metric. n = ldy is the unit normal to the boundary and

∇ is the covariant

derivative defined with the unperturbed bulk metric. γ

= R

ˆγ

is the unperturbed boundary

metric. It is important to keep track of all the boundary terms arising from integration by
parts. Evaluating on shell gives

¯l

2π ¯

dy sinh

−

¯l

16π ¯

4¯l

∂

−

coth y

¯l

. (3.25)

where we are now raising and lowering with ˆ

. The Gibbons-Hawking term is

−

¯l

2π ¯

sinh

cosh y

−

8¯l

∂

(3.26)

The first counter term is

8π ˆ

G¯l

√

3¯l

8π ¯

sinh

−

4¯l

(3.27)

The second counter term is

¯l

32π ¯

√

γR

¯l

32π ¯

12 sinh

−

¯l

sinh

4¯l

sinh

∇

. (3.28)

Thus with only two counter terms we would have

CF T

Ω

8π

log

¯l −

¯l

16π ¯





−

4¯l

∂

¯l





3
2

−

1 +

¯l





¯l

−

8¯l

∇

(3.29)

Ω

is the area of a unit four-sphere and we have used equation 3.9. The expansion of ∂

y = y

is obtained from

∂

)

(p)

(x)

γh

(p)

)

(3.30)

and

)

= 2 +

¯l

(p + 1)(p + 2) + p(p + 1)(p + 2)(p + 3)

¯l

log(¯l/R) +

¯l

+ 2p

− 5p

− 10p − 2 − p(p + 1)(p + 2)(p + 3)(ψ(1) + ψ(2) − ψ(p/2 + 2) − ψ(p/2 + 5/2))

¯l

log(¯l/R)

(3.31)

The psi function is defined by ψ(z) = Γ

(z)/Γ(z). Substituting into the action we find that the

divergences as ¯l

→ 0 cancel at order R

/¯l

and R

/¯l

. The term of order ¯l

in the above

expansion makes a contribution to the finite part of the action (along with a term from the
square root in equation 3.29):

CF T

Ω

8π

log

¯l

(3.32)

256π

γh

(p)

)

2p(p + 1)(p + 2)(p + 3) log(¯l/R) + Ψ(p)

where

Ψ(p) = p(p + 1)(p + 2)(p + 3) [ψ(p/2 + 5/2) + ψ(p/2 + 2)

− ψ(2) − ψ(1)]

+ 2p

− 5p

− 10p − 6.

(3.33)

To cancel the logarithmic divergences as ¯l

→ 0, we have to introduce a length scale ρ defined

by ¯l = ρ and add a counter term proportional to log to cancel the divergence as tends to
zero. The counter term is

−

¯l

64π ¯

log

√

−

1
3

−

¯l

64π ¯

log

−12 +

−

3
2

∇

1
4

∇

. (3.34)

This term does indeed cancel the logarithmic divergence, leaving us with

CF T

Ω

8π

log

(3.35)

256π

γh

(p)

)

(2p(p + 1)(p + 2)(p + 3) log(ρ/R) + Ψ(p))

Note that varying W

CF T

twice with respect to h

yields the expression for the transverse

traceless part of the correlator

(x)T

)

i on a round four sphere. At large p, this behaves

like p

log p, as expected from the flat space result [21]. In fact this correlator can be determined

in closed form solely from the trace anomaly and symmetry considerations

. However, we shall

be be interested in calculating cosmologically observable effects, for which our mode expansion
is more useful.

3.3

The Total Action.

Recall that our five dimensional action is

S = S

+ S

+ 2S

+ W

CF T

(3.36)

In order to calculated correlators of the metric, we need to evaluate the path integral

Z[h] =

∪B

d[δg] exp(

−S)

(3.37)

= exp(

−2S

+ h]

− W

CF T

+ h])

d[δg] exp(

−S

+ δg]

− S

+ δg])

Here g

and h

refer to the unperturbed background metrics in the bulk and on the wall

respectively and h denotes the metric perturbation on the wall. Many of the terms required
here can be obtained from results in the previous section by simply replacing ¯l and ¯

G with l

and G. For example, from equation 3.27 we obtain

+ h] =

8πG

sinh

−

(3.38)

where y

is defined by R = l sinh y

. The path integral over δg is performed by splitting it into

a classical and quantum part:

δg = h + h

(3.39)

where the boundary perturbation h is extended into the bulk using the linearized Einstein
equations and the requirement of finite Euclidean action, i.e., h is given in the bulk by equation
3.21. h

denotes a quantum fluctuation that vanishes at the domain wall. The gravitational

action splits into separate contributions from the classical and quantum parts:

+ S

= S

[h] + S

(3.40)

where S

can be read off from equations 3.25 and 3.26 as

−

Ω

2πG

dy sinh

cosh

16πG

∂

coth y

, (3.41)

Note that S

cannot be converted to a surface term since h

does not satisfy the Einstein

equations. We shall not need the explicit form for S

since the path integral over h

just

contributes a factor of some determinant Z

to Z[h]. We obtain

Z[h] = Z

exp(

−2S

+ h]

− 2S

+ h]

− W

CF T

+ h]).

(3.42)

See [34] for a general discussion of such correlators on maximally symmetric spaces.

The exponent is given by

+ 2S

+ W

CF T

−

Ω

πG

dy sinh

y cosh

y +

3Ω

4πGl

Ω

8π

log

γh

(p)

)

32πG

)

+ 4 coth y

− 6

256π

sinh

(2p(p + 1)(p + 2)(p + 3) log(ρ/R) + Ψ(p))

(3.43)

We have kept the unperturbed action in order to demonstrate how the conformal anomaly

arises: it is simply the coefficient of the log(R/ρ) term divided by the area Ω

of the sphere.

If we set the metric perturbation to zero and vary R in equation 3.43 (using R = l sinh y

) then

we reproduce equation 3.7.

Having calculated R, we can now choose a convenient value for the renormalization scale

ρ. If we were dealing purely with the CFT then we could keep ρ arbitrary. However, since
the third counter term (equation 3.34) involves the square of the Weyl tensor (the integrand is
proportional to the difference of the Euler density and the square of the Weyl tensor), we can
expect pathologies to arise if this term is present when we couple the CFT to gravity. In other
words, when coupled to gravity, different choices of ρ lead to different theories. We shall choose
the value ρ = R so that the third counter term exactly cancels the divergence in the CFT, with
no finite remainder and hence no residual curvature squared terms in the action.

The (Euclidean) graviton correlator can be read off from the action as

(x)h

)

i =

128π

∞

p=2

(p)

iji

(x, x

)F (p, y

)

−1

(3.44)

where we have eliminated l

/G using equation 3.7. The function F (p, y

) is given by

F (p, y

) = e

sinh y

)

+ 4 coth y

− 6

+ Ψ(p),

(3.45)

and the bitensor W

(p)

iji

(x, x

) is defined as

(p)

iji

(x, x

) =

k,l,m,...

(p)

(x)H

(p)

(3.46)

with the sum running over all the suppressed labels k, l, m, . . . of the tensor harmonics.

The appearance of N

in the denominator in equation 3.44 suggests that the CFT suppresses

metric perturbations on all scales. This is misleading because R also depends on N. The function
F (p, y

) has the following limiting forms for large and small radius:

lim

→∞

F (p, y

) = Ψ(p) + p

+ 3p + 6,

(3.47)

lim

→0

F (p, y

) = Ψ(p) + p + 6.

(3.48)

F (p, y

) has poles at p =

−4, −5, −6, . . . with zeros between each pair of negative integers start-

ing at

−3, −4. When we analytically continue to Lorentzian signature, we shall be particularly

interested in zeros lying in the range p

≥ −3/2. There is one such zero exactly at p = 0, another

near p = 0 and a third near p =

−3/2. For large radius, these extra zeros are at p ≈ −0.054

and p

≈ −1.48 while for small radius they are at p ≈ 0.094 and p ≈ −1.60. For intermediate

radius they lie between these values, with the zeros crossing through

−3/2 and 0 at y

≈ 0.632

and y

≈ 1.32 respectively.

3.4

Comparison With Four Dimensional Gravity.

We discussed in section 2 how the RS scenario reprodues the predictions of four dimensional
gravity when the effects of matter on the domain wall dominates the effects of the RS CFT. In
our case we have a CFT on the domain wall. This has action proportional to N

. The RS CFT

is a similar CFT (but with a cut-off) and therefore has action proportional to N

. Hence we

can neglect it when N

. The logarithmic counterterm is also proportional to N

and

therefore also negligible. We therefore expect the predictions of four dimensional gravity to be
recovered when N

. We shall now demonstrate this explicitly.

First consider the radius R of the domain wall given by equation 3.7. It is convenient to

write this in terms of the rank N

of the RS CFT (given by l

/G = 2N

/π)

+ 1 =

16N

(3.49)

If we assume N

1 then the solution is

√

1 +

O(N

)

(3.50)

Note that this implies R

l, i.e., the domain wall is large compared with the anti-de Sitter

length scale.

Now let’s turn to a four dimensional description in which we are considering a four sphere

with no interior. The only matter present is the CFT. The metric is simply

= R

(3.51)

where R

remains to be determined. The action is the four dimensional Einstein-Hilbert ac-

tion (without cosmological constant) together with W

CF T

. There is no Gibbons-Hawking term

because there is no boundary. Without a metric perturbation, the action is simply

S =

−

16πG

√

γR + W

CF T

−

3Ω

4πG

Ω

8π

log

(3.52)

where G

is the four dimensional Newton constant. We want to calculate the value of R

so we

can’t choose ρ = R

yet. Varying R

gives

4π

(3.53)

and N is large hence R

is much greater than the four dimensional Planck length. Substituting

= G

/l, this reproduces the leading order value for R found above from the five dimensional

calculation.

We can now go further and include the metric perturbation. The perturbed four dimensional

Einstein-Hilbert action is

(4)

−

16πG

12R

−

∇

(3.54)

Adding the perturbed CFT gives

S =

−

Ω

16π

Ω

8π

log

γh

(p)

)

64πG

+ 3p + 6)

256π

(2p(p + 1)(p + 2)(p + 3) log(ρ/R

) + Ψ(p))

(3.55)

Setting ρ = R

, we find that the graviton correlator for a four dimensional universe containing

the CFT is

(x)h

)

i = 8N

∞

p=2

(p)

iji

(x, x

)

+ 3p + 6 + Ψ(p)

−1

(3.56)

This can be compared with the expression obtained from the five dimensional calculation, which
can be written

(x)h

)

i =

1 +

O(N

)

∞

p=2

(p)

iji

(x, x

)

+ 3p + 6 + Ψ(p)

(3.57)

+ 4p(p + 1)(p + 2)(p + 3)(N

) log(N

/N) +

O(N

)

−1

We have expanded in terms of

πl

(3.58)

The four and five dimensional expressions clearly agree (for G

= G/l) when N

, i.e.,

l. There are corrections of order (N

) log(N

/N ) coming from the RS CFT and the

logarithmic counter term. In fact, these corrections can be absorbed into the renormalization
of the CFT on the domain wall if, instead of choosing ρ = R, we choose

ρ = R

−

log(N

/N )

(3.59)

The corrections to the four dimensional expression are then of order N

. We shall not give

these correction terms explicitly although they are easily obtained from the exact result 3.44.

3.5

Lorentzian Correlator.

In this subsection we shall show how the Euclidean correlator calculated above is analytically
continued to give a correlator for Lorentzian signature. We have put many of the details in

Appendix B but the analysis is still rather technical so the reader may wish to skip to the final
result, which is given in equation 3.66. The techniques used here were developed in [35, 36, 37].

Let us first introduce a new label p

= i(p + 3/2), so that on the four sphere

∇

)

= λ

)

(3.60)

where p

= 7i/2, 9i/2, ... and

= (p

+ 17/4).

(3.61)

Recall that there are extra labels on the tensor harmonics that we have suppressed. The
set of rank-two tensor eigenmodes on S

forms a representation of the symmetry group of

the manifold. Hence the sum (equation B.2) of the degenerate eigenfunctions with eigenvalue
λ

defines a maximally symmetric bitensor W

) i

(µ(Ω, Ω

)), where µ(Ω, Ω

) is the distance

along the shortest geodesic between the points with polar angles Ω and Ω

. The expression of

the bitensor in terms of a set of fundamental bitensors with µ-dependent coefficient functions
together with the relation between the bitensors on S

and Lorentzian de Sitter space are

obtained in Appendix B.

The motivation for the unusual labelling is that, as demonstrated in Appendix B, in terms

of the label p

the bitensor on S

has exactly the same formal expression as the corresponding

bitensor on Lorentzian de Sitter space. This property will enable us to analytically continue the
Euclidean correlator into the Lorentzian region without Fourier decomposing it. In other words,
instead of imposing by hand a prescription for the vacuum state of the graviton on each mode
separately and propagating the individual modes into the Lorentzian region, we compute the
two-point tensor correlator in real space, directly from the no boundary path integral. Since the
path integral unambiguously specifies the allowed fluctuation modes as those which vanish at
the origin of the instanton (see discussion in subsection 3.2), this automatically gives a unique
Euclidean correlator. The technical advantage of our method is that dealing directly with the
real space correlator makes the derivation independent of the gauge ambiguities involved in the
mode decomposition [37].

We begin by continuing the graviton correlator (equation 3.44) obtained via the five di-

mensional calculation. The analytic continuation of the correlator for four dimensional gravity
(equation 3.56) is completely analogous. In terms of the new label p

, the Euclidean correlator

3.44 between two points on the wall is given by

(Ω)h

(Ω

)

i =

128π

∞

=7i/2

)

iji

(µ)G(p

, y

)

−1

(3.62)

where

G(p

, y

) = F (

−ip

− 3/2, y

)

(3.63)

= e

sinh y

)

+ 4 coth y

− 6

− 4ip

+ p

− 5ip

− 63/16

+ (p

+ 1/4)(p

+ 9/4)[ψ(

−ip

/2 + 5/4) + ψ(

−ip

/2 + 7/4)

− ψ(1) − ψ(2)]

with g

(y) = Q

−ip

−1/2

(coth y), which follows from eq. 3.19. The function G(p

, y

) is real and

positive for all values of p

in the sum and for arbitrary y

≥ 0.

We have the Euclidean correlator defined as an infinite sum. However, the eigenspace of

the Laplacian on de Sitter space suggests that the Lorentzian propagator is most naturally
expressed as an integral over real p

. We must therefore first analytically continue our result

from imaginary to real p

. The coefficient G(p

, y

)

−1

of the bitensor is analytic in the upper half

complex p

-plane, apart from three simple poles on the imaginary axis. One of them is always

at p

= 3i/2, regardless of the radius of the sphere. Let the position of the remaining two poles

be written p

= iΛ

). If we take the radius of the domain wall to be large compared with the

AdS scale (which is necessary for corrections to four dimensional Einstein gravity to be small)
then

0 < Λ

≤ 3/2, with Λ

∼ 0 and Λ

∼ 3/2. Since G(p

, y

) is real on the imaginary

-axis, the residues at these poles are purely imaginary. In order to extend the correlator into

the complex p

-plane, we must also understand the continuation of the bitensor itself. As shown

in Appendix B, the condition of regularity at opposite points on the four sphere imposed by the
completeness relation (equation B.4) is sufficient to uniquely specify the analytic continuation
of W

)

iji

(µ) into the complex p

-plane. The extended bitensor is defined by equations B.5, B.8

and B.11.

Now we are able to write the sum in equation 3.62 as an integral along a contour

encircling

the points p

= 7i/2, 9i/2, ..ni/2, where n tends to infinity. This yields

(Ω)h

(Ω

)

i =

−i64π

tanh p

πW

)

iji

(µ)G(p

, y

)

−1

(3.64)

Since we know the analytic properties of the integrand in the upper half complex p

-plane,

we can distort the contour for the p

integral to run along the real axis. At large imaginary p

the integrand decays and the contribution vanishes in the large n limit. However as we deform
the contour towards the real axis, we encounter three extra poles in the cosh p

π factor, the pole

at p

= 3i/2 becoming a double pole due to the simple zero of G(p

, y

). In addition, we have

to take in account the two poles of G(p

, y

)

−1

at p

= iΛ

For the p

= 5i/2 pole, it follows from the normalization of the tensor harmonics that

(5i/2)

iji

= 0. Indirectly, this is a consequence of the fact that spin-2 perturbations do not have a

dipole or monopole component. The meaning of the remaining two poles of the tanh p

π factor

has been extensively discussed in [37], where the continuation is described of the two-point
tensor fluctuation correlator from a four dimensional O(5) instanton into open de Sitter space.
They represent non-physical contributions to the graviton propagator, arising from the different
nature of tensor harmonics on S

and on Lorentzian de Sitter space. In fact, a degeneracy

appears between p

= 3i/2 and p

= i/2 tensor harmonics and respectively p

= 5i/2 vector

harmonics and p

= 5i/2 scalar harmonics on S

. More precisely, the tensor harmonics that

constitute the bitensors W

iji

(3i/2)

and W

iji

(i/2)

can be constructed from a vector (scalar) quantity.

Consequently, the contribution to the correlator from the former pole is pure gauge, while
the latter eigenmode should really be treated as a scalar perturbation, using the perturbed
scalar action. Henceforth we shall exclude them from the tensor spectrum. This leaves us with
the poles of G(p

, y

) at p

= iΛ

. If we deform the contour towards the real axis, we must

compensate for them by subtracting their residues from the integral. We will see that these
residues correspond to discrete “supercurvature” modes in the Lorentzian tensor correlator.

If we decrease the radius of the domain wall, then the poles move away from each other. Their behaviour

follows from the discussion below equations 3.47 and 3.48. For y

≤ 0.632, Λ

becomes slightly smaller than

zero while for y

≤ 1.32, Λ

becomes slightly greater than 3/2.

The contribution from the closing of the contour in the upper half p

-plane vanishes. Hence

our final result for the Euclidean correlator reads

(Ω)h

(Ω

)

i =

−i64π

∞

−∞

tanh p

πW

)

iji

(µ)G(p

, y

)

−1

+2π

k=1

tan Λ

πW

(iΛ

)

iji

(µ)Res(G(p

, y

)

−1

; iΛ

)

(3.65)

The analytic continuation from a four sphere into Lorentzian closed de Sitter space is given

by setting the polar angle Ω = π/2

− it. Without loss of generality we may take µ = Ω, and µ

then continues to π/2

− it. We then obtain the correlator in de Sitter space where one point

has been chosen as the origin of the time coordinate.

The continuation of the bitensor W

)

iji

(µ) is given in Appendix B. An extra subtlety arises

if one wants to identify the continued bitensor with the usual sum of tensor harmonics on de
Sitter space. It turns out that in order to do so, one must extract a factor ie

pπ

/ sinh p

π from

its coefficient functions

. We denote the final form of the bitensor by W

L(p

)

iji

(µ(x, x

)), which

is defined in the Appendix, equations B.5, B.8 and B.16.

The extra factor ie

pπ

/ sinh p

π combines with the factor

−i tanh p

π in the integrand to

/ cosh p

π. Furthermore, since G(

−p

, y

) = ¯

G(p

, y

), we can rewrite the correlator as an in-

tegral from 0 to

∞. We finally obtain the Lorentzian tensor Feynman (time-ordered) correlator,

(x)h

)

i =

128π

∞

tanh p

πW

L(p

)

iji

(µ)

<(G(p

, y

)

−1

)

+π

k=1

tan Λ

πW

L(iΛ

)

iji

(µ)Res(G(p

, y

)

−1

; iΛ

)

128π

∞

L(p

)

iji

(µ)

<(G(p

, y

)

−1

)

−π

k=1

L(iΛ

)

iji

(µ)Res(G(p

, y

)

−1

; iΛ

)

(3.66)

In this integral the bitensor W

L(p

)

iji

(µ(x, x

)) may be written as the sum of the degenerate

rank-two tensor harmonics on closed de Sitter space with eigenvalue λ

= (p

+ 17/4) of

the Laplacian. Note that the normalization factor ˜

= p

(4p

+ 25)/48π

of the bitensor is

imaginary at p

= iΛ

and the residues of G

−1

are also imaginary, so the quantities in square

brackets are all real. Both integrands in equation 3.66 vanish as p

→ 0, so the correlator is

well-behaved in the infrared.

For cosmological applications, one is usually interested in the expectation of some quantity

squared, like the microwave background multipole moments. For this purpose, all that matters
is the symmetrized correlator, which is just the real part of the Feynman correlator.

The underlying reason is that there exist two independent bitensors of the form defined by equations B.5 and

B.8. Under the integral in the Lorentzian correlator, they are related by the factor ie

pπ

/ sinh p

π. It follows from

the continuation of the completeness relation (equation B.4) that the sum of degenerate tensor harmonics on
de Sitter space equals the second independent bitensor, rather then the bitensor that we obtain by continuation
from S

. Therefore, in order to express the Lorentzian two-point tensor correlator in terms of tensor harmonics,

we must extract this factor from the bitensor. We refer the interested reader to the Appendix for the details.

Gravitational waves provide an extra source of time-dependence in the background in which

the cosmic microwave background photons propagate. In particular, the contribution of grav-
itational waves to the CMB anisotropy is given by the integral in the Sachs-Wolfe formula,
which is basically the integral along the photon trajectory of the time derivative of the tensor
perturbation. Hence the resulting microwave multipole moments

can be directly determined

from the graviton correlator.

We can therefore understand the effect of the strongly coupled CFT on the microwave

fluctuation spectrum by comparing our result 3.66 with the transverse traceless part of the
graviton propagator in four-dimensional de Sitter spacetime [41]. On the four-sphere, this is
easily obtained by varying the Einstein-Hilbert action with a cosmological constant. In terms
of the bitensor, this yields

(Ω)h

(Ω

)

i = 32πG

∞

=7i/2

)

iji

(µ(Ω, Ω

))

− 2

(3.67)

which continues to

(x)h

)

i = 32πG

∞

− 2

L(p

)

iji

(µ(x, x

)).

(3.68)

This can be compared with equation 3.66. Note that (apart from the pole at p

= 3i/2 corre-

sponding to the gauge mode mentioned before) there are no supercurvature modes. We defer
a detailed discussion of the effect of the CFT on the tensor perturbation spectrum in de Sitter
space to the next section.

Conclusion

We have studied a Randall-Sundrum cosmological scenario consisting of a domain wall in anti-de
Sitter space with a large N conformal field theory living on the wall. The confomal anomaly of
the CFT provides an effective tension which leads to a de Sitter geometry for the domain wall.
We have computed the spectrum of quantum mechanical vacuum fluctuations of the graviton
field on the domain wall, according to Euclidean no boundary initial conditions. The Euclidean
path integral unambiguously specifies the tensor correlator with no additional assumptions.
This is the first calculation of quantum fluctuations for RS cosmology.

In the usual inflationary models, one considers the classical action for a single scalar field.

In that context, it is consistent to neglect quantum matter loops, on the grounds that they are
small. On the other hand, in this paper we have studied a strongly coupled large N CFT living
on the domain wall, for which quantum loops of matter are important. By using the AdS/CFT
correspondence, we have performed a fully quantum mechanical treatment of this CFT. The
most notable effect of the large N CFT on the tensor spectrum is that it suppresses small scale
fluctuations on the microwave sky. It can be seen from equation 3.66 that the CFT yields a
(p

ln p

)

−1

behaviour for the graviton propagator at large p

(in agreement with the flat space

results of [40]), instead of the usual p

0−2

falloff (equation 3.68). In other words, quantum loops

of the CFT give spacetime a rigidity that strongly suppresses metric fluctuations on small scales.
Note that this is true independently of how the de Sitter geometry arises, i.e. it is also true for

four dimensional Einstein gravity. In addition, the coupling of the CFT to tensor perturbatons
gives rise to two additional discrete modes in the tensor spectrum. Although this is a novel
feature in the context of inflationary tensor perturbations, it is not surprising. In conventional
open inflationary scenarios for instance, the coupling of scalar field fluctuations with scalar
metric perturbations introduces a supercurvature mode with an eigenvalue of the Laplacian
close to the discrete de Sitter gauge mode [42, 35]. The former discrete mode at p

= iΛ

∼ 3i/2

in equation 3.66 is nothing else than the analogue of this well known supercurvature mode
in the scalar fluctuation spectrum. The second mode has an eigenvalue p

= iΛ

∼ 0. Its

interpretation is less clear, but it is clearly an effect of the matter on the domain wall. However
it hardly contributes to the correlator because tan Λ

π is very small.

The effect of the CFT on large scales is more difficult to quantify because of the complicated

-dependence of the tensor correlator (equation 3.66) in the low-p

regime. Generally speaking,

however, long-wavelength tensor correlations in closed (or open) models for inflation are very
sensitive to the details of the underlying theory, as well as to the boundary conditions at
the instanton. Since tensor fluctuations do give a substantial contribution to the large scale
CMB anisotropies, this may provide an additional way to observationally distinguish different
inflationary scenarios [38].

Most matter fields can be expected to behave like a CFT at small scales. Furthermore,

fundamental theories such as string theory predict the existence of a large number of matter
fields. Therefore, our results based on a quantum treatment of a large N CFT may be accurate
at small scales for any matter. If this is the case then our result shows that tensor perturbations
at small angular scales are much smaller than predicted by calculations that neglect quantum
effects of matter fields.

Acknowledgments

It is a pleasure to thank Steven Gratton, Hugh Osborn and Neil Turok for useful discussions.

Choice of Gauge

This appendix demonstrates how a metric perturbation on the boundary of a ball of AdS is
decomposed into vector, scalar and tensor components.

Consider a ball of perturbed AdS with a spherical boundary. Let ¯l be the AdS length scale.

Gaussian normal coordinates are introduced by defining ¯ly to be the geodesic distance of a point
from the origin. The surfaces of constant y are spheres on which we introduce coordinates x

In these coordinates the metric takes the form

= ¯l

(dy

+ sinh

yˆ

(x)dx

) + h

(y, x)dx

(A.1)

The ball of AdS has been perturbed, so the boundary will be at a position y = y

+ ξ(x).

Let the induced metric perturbation on the boundary be ˆh

(x). This can be decomposed

into scalar, vector and tensor perturbations with respect to the round metric on the sphere [39]:

ˆh

(x) = ˆ

+ 2 ˆ

∇

+ ˆ

∇

φ + ˆ

ψ,

(A.2)

where we use hats to denote quantities defined on the sphere (i.e. quantities that depend only on
x). ˆ

is a transverse traceless tensor on the sphere and ˆ

is a transverse vector on the sphere.

φ and ˆ

ψ are scalars on the sphere. ˆ

and ˆ

φ can be gauged away by infinitesimal coordinate

transformations on the sphere of the form x

= ˜

− η

(˜

− ∂

η(˜

x) where η

is transverse.

Therefore we shall assume that ˆ

χ and ˆ

φ vanish. Note that it is not possible to gauge away ˆ

or ξ. This paper only deals with tensor perturbations so we shall assume that the scalars ˆ

and ξ are vanishing. The induced metric perturbation is then transverse and traceless and can
be extended into the bulk as described in section 3. The scalars will be discussed in our next
paper.

Maximally Symmetric Bitensors.

A maximally symmetric bitensor T is one for which σ

∗

T = 0 for any isometry σ of the maxi-

mally symmetric manifold. Any maximally symmetric bitensor may be expanded in terms of a
complete set of fundamental maximally symmetric bitensors with the correct index symmetries.
For instance

iji

= t

(µ)g

+ t

(µ)n

j)(i

)

+ t

(µ)

+ g

(µ)n

+ t

(µ)

+ n

(B.1)

The coefficient functions t

(µ) depend only on the distance µ(Ω, Ω

) along the shortest geodesic

from the point Ω to the point Ω

. n

(Ω, Ω

) and n

(Ω, Ω

) are unit tangent vectors to the

geodesics joining Ω and Ω

and g

(Ω, Ω

) is the parallel propagator along the geodesic, i.e.,

is the vector at Ω

obtained by parallel transport of V

along the geodesic from Ω to Ω

[43].

The set of tensor eigenmodes on S

(or on de Sitter space) forms a representation of the

symmetry group of the manifold. It follows in particular that their sum over the parity states
P = {e, o} and the quantum numbers k, l and m on the three sphere defines a maximally
symmetric bitensor on S

(or dS space) [43]:

) i

(µ) =

Pklm

)ij

Pklm

(Ω)q

)

Pklm

(Ω

)

∗

(B.2)

On S

the label p

takes the value 7i/2, 9i/2, ... It is related to a real label p by p

= i(p + 3/2).

The ranges of the other labels are then 0

≤ k ≤ p, 0 ≤ l ≤ k and −l ≤ m ≤ l. On de Sitter

space there is a continuum of eigenvalues p

∈ [0, ∞). We will assume from now on that the

eigenmodes are normalized by the condition

√

γd

Ωq

)ij

Pklm

)

∗

= δ

(B.3)

The completeness relation on the four sphere may then be written as

−

1
2

(Ω

− Ω

) =

∞

=7i/2

) i

(µ(Ω, Ω

)).

(B.4)

Explicit formulae for the components of these tensors may be found in [33]. In this Appendix
we will determine W

)

iji

(µ) simultaneously on the four sphere and de Sitter space. The con-

struction of the analogous bitensor on S

and H

is given in [44] and their relation is described

in [37].

The bitensor W

) i

(µ) has some additional properties arising from its construction in

terms of the transverse and traceless tensor harmonics q

(p)

Pklm

. The tracelessness of W

)

iji

allows one to eliminate two of the coefficient functions in equation B.1. It may then be written
as

)

iji

(µ) = w

)

− 4n

i h

− 4n

+ w

)

j)(i

)

+ 4n

)

+ g

− 2n

+ 8n

(B.5)

This expression is traceless on either the index pair ij or i

. The requirement that the bitensor

be transverse

∇

)

iji

= 0 and the eigenvalue condition (

∇

−λ

iji

)

= 0 impose additional

constraints on the remaining coefficient functions w

)

(µ). To solve these constraint equations it

is convenient to introduce the new variables on S

(in de Sitter space, µ is replaced by π/2

− i˜µ)







α(µ) = w

(p)
1

(µ) +

2
3

(p)
3

(µ)

β(µ) =

(λ

+8) sin µ

dα(µ)

dµ

(B.6)

In terms of a new argument z = cos

(µ/2) (or its continuation on de Sitter space) the transver-

sality and eigenvalue conditions imply for α(z)

z(1

− z)

α(z)

+ [4

− 8z]

dα(z)

= (λ

+ 8)α(z)

(B.7)

and then for the coefficient functions











−

6
5

[(λ

+ 28)z(1

− z) − 45/6] α(z) +

[(λ

+ 8)z(1

− z)(1 − 2z)] β(z)

9
5

(λ

+ 28)z(1

− z) +

− z) −

α(z)

−

[(λ

+ 8)z(1

− z)(4 − 3z)] β(z)

9
5

[(λ

+ 28)z(1

− z) − 40/6] α(z) −

[(λ

+ 8)z(1

− z)(1 − 2z)] β(z)

(B.8)

with λ

= (p

+ 17/4).

Notice that equation B.7 is precisely the hypergeometric differential equation, which has a

pair of independent solutions α(z) and α(1

− z) where

α(z) = Q

(7/2 + ip

, 7/2

− ip

, 4, z)

(B.9)

is a constant. The solution for β(z) follows from equation B.6 and is given by

β(z) = Q

(9/2

− ip

, 9/2 + ip

, 5, z).

(B.10)

We will determine below which solution corresponds to the bitensor defined by B.2.

Our discussion so far applies to either S

or de Sitter space. We now specialize to the case of

and will later obtain results for de Sitter space by analytic continuation. The hypergeometric

functions on S

may be expressed in terms of Legendre polynomials in cos µ (eq. [15.4.19] in

[45]),

(

α(µ) = Q

Γ(4)2

(sin µ)

−3

−1/2+ip

(

− cos µ),

β(µ) = Q

Γ(5)2

(sin µ)

−4

−1/2+ip

(

− cos µ).

(B.11)

The solutions for α(z) and β(z) are singular at z = 1 (i.e. for coincident points on S

) for generic

values of p

. However, for the values of p

corresponding to the eigenvalues of the Laplacian on

, they are regular everywhere on S

. Similarly, α(1

− z) and β(1 − z) are generically singular

for antipodal points on S

and regular for these special values of p

. For these special values,

α(z) and α(1

−z) are no longer linearly independent but related by a factor of (−1)

(n+1)/2

where

n =

−2ip

= 7, 9, 11, . . .. This follows from the relation (eq.[8.2.3] in [45])

(

−z) = e

iνπ

(z)

−

−iµπ

sin[π(ν + µ)]Q

(z),

(B.12)

where the second term vanishes for p

= 7i/2, 9i/2, .... In fact, the hypergeometric series termi-

nates for these values of p

and the hypergeometric functions reduce to Gegenbauer polynomials

(7/2)

−7/2

− 2z). We have a choice between using α(z) and α(1 − z) in the bitensor for these

values of p

. However, to obtain the Lorentzian correlator, we had to express the discrete sum

3.62 as a contour integral. Since the Euclidean correlator obeys a differential equation with a
delta function source at µ = 0, we must maintain regularity of the integrand at µ = π when
extending the bitensor in the complex p

-plane. In other words, for generic p

, we need to work

with the solution α(z), rather then α(1

− z). We shall therefore choose α(z), since this is the

solution that we will analytically continue.

The above conditions leave the overall normalisation of the bitensor undetermined. To fix

the normalisation constant Q

, consider the biscalar quantity

)

iji

(µ) = 12w

)

− 6w

)

+ 24w

)

(B.13)

In the coincident limit Ω

→ Ω

and z

→ 1 this yields

) ij

(Ω, Ω) =

Pklm

)

Pklm

(Ω)q

)

Plm ij

(Ω)

∗

−72α(1).

(B.14)

Since F (0) = 1 we have α(1) = Q

(

−1)

(1+n)/2

. By integrating over the four-sphere and using the

normalization condition B.3 on the tensor harmonics one obtains, for n =

−2ip

= 7, 9, 11, . . .

(4p

+ 25)

48π

(

−1)

(1+n)/2

(4p

+ 25)

48π

sinh p

(B.15)

We conclude that the properties of the bitensor appearing in the tensor correlator completely

determine its form. Notice that in terms of the label p

we have obtained a unified functional

description of the bitensor on S

and de Sitter space. However, its explicit form is very different

in the two cases because the label p

takes on different values. It is precisely this description that

has enabled us in section 3 to analytically continue the correlator from the Euclidean instanton
into de Sitter space without Fourier decomposing it. We shall conclude this Appendix by
describing in detail the subtleties of this analytic continuation at the level of the bitensor.

To perform the continuation to de Sitter space we note that the geodesic separation µ on

continues to π/2

− it, so z =

1
2

(1 + i sinh t) on de Sitter space. The continuation of the

hypergeometric functions (B.11) yields

(

α(z) = Γ(4)2

(cosh t)

−3

−1/2+ip

(

−i sinh t),

β(z) = Γ(5)2

(cosh t)

−4

−1/2+ip

(

−i sinh t).

(B.16)

However, an extra subtlety arises if one wants to identify the continued bitensor with the

usual sum of tensor harmonics on de Sitter space. In particular, in order for the bitensor
to correspond to the usual sum of rank-two tensor harmonics on the real p

-axis, one must

choose the second solution α(1

− z) to the hypergeometric equation, rather then α(z) that

enters in the continued bitensor. This is easily seen by performing the continuation on the
completeness relation (equation B.4), which should continue to an integral over p

from 0 to

∞

of the Lorentzian bitensor, defined as the sum (B.2) over the degenerate tensor harmonics on de
Sitter space. Writing (B.4) as a contour integral and continuing to Lorentzian de Sitter space
yields

−

1
2

− x

) =

∞

−∞

tanh p

πW

) i

(µ(x, x

)).

(B.17)

Clearly this is not the correct completeness relation according to the equivalent definition (B.2)
of the bitensor on de Sitter space. But let us relate the continued bitensor in (B.17) to the
independent bitensor in which the solutions α(1

− z) enter. This can be done by applying

(B.12) to the Legendre polynomials in (B.16). By closing the contour in the upper half p

-plane,

one sees there is no contribution to the integral (and indeed to the tensor correlator!) from
the second term in equation B.12, because its prefactor cancels the cosh

−1

π)-factor in (B.17),

making the integrand analytic in the upper half p

-plane (up to gauge modes). Hence, under the

integral both solutions are simply related by the factor ie

pπ

. In addition one needs to extract

the sinh

−1

π-factor

from Q

. The completeness relation then becomes,

−

1
2

− x

) =

∞

L(p

) i

(µ(x, x

)),

(B.18)

and the hypergeometric functions α(1

− z) and β(1 − z) that constitute the final bitensor

L(p

) i

(µ(x, x

)) are given by

(

α(1

− z) = ˜

Γ(4)2

(cosh t)

−3

−1/2+ip

(i sinh t),

β(1

− z) = ˜

Γ(5)2

(cosh t)

−4

−1/2+ip

(i sinh t),

(B.19)

with ˜

= p

(4p

+ 25)/48π

On the real p

-axis, W

L(p

)

iji

(µ) equals the sum (B.2) of the degenerate rank-two tensor har-

monics on closed de Sitter space with eigenvalue λ

= (p

+ 17/4) of the Laplacian.

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